The figure shows the block diagram of a typical modem circa 1985. The illustrated device is an external modem, meaning that it sits in a box beside the computer and has an RS-232 serial connection to the computer. It also requires its own power supply.
The device contains a few analog components which behave broadly like a standard telephone, but most of it is digital. A relay is used to connect the device to the line and its contacts mirror the `off-hook' switch which is part of every telephone. It connects a transformer across the line. The relay and transformer provide isolation of the computer ground signal from the line voltages. Similarly the ringing detector often uses a opto-coupler to provide isolation. Clearly, these analog aspects of the design are particular to a modem and are designed by a telephone expert.
Modems from the 1960's implemented everything in analog circuitry since microprocessors and DSP were not available. In 1985, two microprocessors were often used.
Note that the non-volatile RAM required (and still does) a special manufacturing processing step and so is not included as a resource on board the single-chip processor. Similarly, the RS-232 drivers need to handle voltages of +/- 12 volts and so these cannot be included on chip without increasing the cost of the rest of the chip by using a fabrication process which can handle these voltages. The NV-RAM is used to store the owner's settings, such as whether to answer an incoming call and what baud rate to attempt a first connection, etc..